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Homework Help: Prove that the series SUM (-1)^n n/p_n converges where p_n are primes

  1. Mar 5, 2007 #1
    1. The problem statement, all variables and given/known data

    Prove that [tex]\sum(-1)^n\frac{n}{p_n}[/tex] converges, where [tex]p_n[/tex] is the nth prime.

    2. Relevant equations

    The sequence [tex]\frac{n}{p_n}[/tex] is definately not monotone if there exists infinitely many twin primes, since [tex]2n-p_n<0[/tex] for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?
     
  2. jcsd
  3. Mar 5, 2007 #2

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    Can you use the prime number theorem? This says that:

    [tex] \lim_{n \rightarrow \infty} \frac{p_n}{n \ln n} = 1 [/tex]
     
  4. Mar 10, 2007 #3
    I can't use it for the series. I can only establish that n/p_n -> 0, which is insufficient for the series. I can't even prove that n/p_n is NOT monotone for large n, unless I assume the twin prime conjecture, for example.
     
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